Multidimensional Summation-by-Parts Operators: General Theory and Application to Simplex Elements

Abstract

Abstract. Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely determined by a basis, so the potential exists to tailor SBP operators to meet different objectives. To date, application of highorder SBP discretizations to multiple dimensions has been limited to tensor product domains. This paper presents a definition for multi-dimensional SBP finite-difference operators that is a natural extension of one-dimensional SBP operators. Theoretical implications of the definition are investigated for the special case of a diagonal norm (mass) matrix. In particular, a diagonal-norm SBP operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial-basis matrix has full rank when evaluated at the cubature nodes. Appropriate simultaneous-approximation terms are developed to impose boundary conditions weakly, and the resulting discretizations are shown to be time stable. Concrete examples of multi-dimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences with spectralelement and spectral-difference methods are discussed. An assembly process is described that builds diagonal-norm SBP operators on a global domain from element-level operators. Numerical results of linear advection on a doubly periodic domain demonstrate the accuracy and time stability of the simplex operators.Key words. summation-by-parts, finite-difference method, unstructured grid, spectral-element method, spectral-difference method, mimetic discretization AMS subject classifications. 65N06, 65M60, 65N121. Introduction. Summation-by-parts (SBP) operators are high-order finitedifference schemes that mimic the symmetry properties of the differential operators they approximate [19]. Respecting such symmetries has important implications; in particular, they enable SBP discretizations that are both time stable and high-order accurate [4,34,27], properties that are essential for robust, long-time simulations of turbulent flows [25,35].Most existing SBP operators are one-dimensional [30,24,31,23] and are applied to multi-dimensional problems using a multi-block tensor-product formulation [32,14,29]. Like other tensor-product methods, the restriction to multi-block grids complicates mesh generation and adaptation, and it limits the geometric complexity that can be considered in practice.The limitations of the tensor-product formulation motivate our interest in generalizing SBP operators to unstructured grids. There are two ways this generalization has been pursued in the literature: 1) construct global SBP operators on an arbitrary distribution of nodes, or; 2) construct SBP operators on reference elements and assemble a global discretization by coupling these smaller elements.The first approach is appealing conceptually, and it is certainly viable for secondor...

Simultaneous Approximation Terms for Multidimensional Summation-by-parts Operators

“…This constraint was imposed, in part, to simplify the construction of pointwise SATs, but it increases the total number of nodes required for the SBP cubature. For example, the quadratic, cubic, and quartic SBP operators for the triangle require 7, 12, and 18 nodes, respectively, rather than the 6, 10, and 15 nodes necessary for a total-degree basis [13]. A similar trend is observed for tetrahedral elements.…”

“…Building on the generalization in [9], we presented an SBP definition in [13] (see also [14]) that is suitable for arbitrary, bounded subdomains with piecewise smooth, orientable boundaries. For diagonal-norm 1 multi-dimensional SBP operators that are exact for polynomials of total degree p, it was shown that the norm and corresponding nodes define a strong cubature rule that is exact for polynomials of degree 2p − 1.…”

“…In [13] we derived SATs for multidimensional diagonal-norm SBP operators and showed that the resulting discretizations are stable for the linear constant-coefficient advection equation. Indeed, for constant-coefficient advection these penalties are the strong-form equivalent of the boundary-integrated numerical flux functions used in [20].…”

“…Indeed, for constant-coefficient advection these penalties are the strong-form equivalent of the boundary-integrated numerical flux functions used in [20]. The SATs described in [13] can theoretically accommodate variable-coefficient advection problems; however, they are not practical for this class of problem because new SBP operators would be needed whenever the variable coefficients change.…”

“…The multi-dimensional SBP operators in [13] were designed to have a unisolvent set of nodes on each face for the appropriate space of polynomials. This constraint was imposed, in part, to simplify the construction of pointwise SATs, but it increases the total number of nodes required for the SBP cubature.…”